A373930 Number of compositions of 7*n-4 into parts 1 and 7.

1, 5, 22, 105, 518, 2555, 12565, 61748, 303470, 1491567, 7331205, 36033501, 177107406, 870496256, 4278555247, 21029425081, 103361226864, 508028305120, 2496997824041, 12272934541014, 60322408298439, 296489232532277, 1457267166329605, 7162579146364783

Offset

1,2

Links

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (8,-21,35,-35,21,-7,1).

Formula

a(n) = A005709(7*n-4).

a(n) = Sum_{k=0..n} binomial(n+2+6*k,n-1-k).

a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

G.f.: x*(1-x)^3/((1-x)^7 - x).

a(n) = n*(1 + n)*(2 + n)*hypergeom([1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6], [4/7, 5/7, 6/7, 8/7, 9/7, 10/7], -6^6/7^7)/6. - Stefano Spezia, Jun 23 2024

Mathematica

a[n_]:=n*(1 + n)*(2 + n)*HypergeometricPFQ[{1-n, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6}, {4/7, 5/7, 6/7, 8/7, 9/7, 10/7}, -6^6/7^7]/6; Array[a, 24] (* Stefano Spezia, Jun 23 2024 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+2+6*k, n-1-k));

Crossrefs

Cf. A099253, A373907, A373928, A373929, A373931, A373932.

Cf. A005709, A369807.

Sequence in context: A296163 A008485 A213684 * A082297 A267241 A162271

Adjacent sequences: A373927 A373928 A373929 * A373931 A373932 A373933

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 23 2024

Status

approved